Relativity of Reconstruction: Observer–Dependent Vacuum Energy in de Sitter Space

Relativity of Reconstruction: Observer–Dependent Vacuum Energy in de Sitter Space Abstract Recent advances in gravitational dressing and quantum reference frames have shown that the algebra of observables associated with horizons acquires a Type II structure, permitting a well-defined notion of generalized entropy. These works characterize the entropy associated with horizons but do not address how an observer should assign expectation values to local energy–momentum observables — the quantities that enter the semiclassical Einstein equation. In this paper we develop the Relativity of Reconstruction (RoR), a framework for defining observer-dependent effective stress tensors in spacetimes with horizons. We construct a class Mholo of admissible coarse-graining maps Fq, characterized by positivity, static-patch covariance, a single geometric scale ℓq∼H−1, and a thermodynamically fixed modular contribution. For any Fq∈ Mholo, we prove a universality theorem: the geometric contribution to ⟨FqTµν⟩scales as H4independently of kernel shape, UV field content, or interactions, while the modular term contributes a universal H2M2 Pl piece fixed by de Sitter thermodynamics. The resulting observer-dependent effective stress tensor takes the form ⟨Tµν(q)⟩=−(C1H4+C2H2M2 Pl)gµν, with Fq→id and Tµν(q)→Tµν in the flat-space limit. We then show that the family {Tµν(q)}defines a (G, X)-structure on the space of observers, with transition maps determined by the coordinates of Fqwithin Mholo. This global “observer atlas” encodes the obstruction to any observer-independent notion of vacuum energy and provides the operator backbone for an observer-relative semiclassical Einstein equation developed in future work. 1 Introduction Quantum field theory assigns to the vacuum a formally divergent energy density of order ρQFT ∼M4 Pl, while cosmological observations infer an effective value ρobs Λ∼H2M2 Pl. The conventional interpretation assumes that both numbers refer to a single, universal vacuum energy, producing the familiar discrepancy of 10120. In this work we pursue a different premise: vacuum energy is not a global scalar but an observer–dependent reconstruction. Each observer q, bounded by a cosmological or acceleration horizon, accesses only a causal subalgebra Aqof the global quantum state ρ. We 1 therefore define the physically meaningful vacuum energy as the reconstructed expectation ρphys vac (q) := (FqT00)ρ,(1) where Fqis an admissible, positive, normalized reconstruction map adapted to the observer’s causal patch. This quantity ρphys vac (q) is not the global ρQFT but the vacuum energy operationally available to that observer. This definition rephrases the cosmological constant mismatch as a category error rather than a failure of quantum field theory or general relativity: ρQFT characterizes the global state, whereas ρobs Λcharacterizes a causal reconstruction. The present framework preserves both theories intact while revising a single assumption: that vacuum energy is globally accessible and gravitationally active for all observers. To clarify the gravitational role of the reconstructed energy, we propose an observer– relative semiclassical Einstein equation Gµνg(q)= 8πG T(q) µν , T(q) µν := Fq(⟨Tµν⟩ρ),(2) in which curvature responds to reconstructed, rather than global, stress–energy. Applied at the level of de Sitter scaling, the interplay of coarse–grained vacuum modes and horizon thermodynamics yields ρeff(q)∼H2M2 Pl,(3) without requiring fine tuning or cancellations. The scaling arises not from renormalization but from reconstruction: ultraviolet contributions suppressed by Fqand the modular (horizon–entropy) term fixed by TdS and SdS. Two central challenges remain open. First, a fully covariant definition of the observer– restricted stress tensor T(q) µν must be given that is compatible with the Bianchi identities. Second, conservation may hold only patchwise, ∇µTµν (q)= 0, or include horizon flux terms consistent with causal inaccessibility. These technical questions are not treated as solved but as a concrete roadmap for subsequent development. In summary, the Relativity of Reconstruction aims not to modify quantum field theory or general relativity but to refine the ontological status of vacuum energy: curvature responds to the energy an observer can reconstruct from within their causal domain, not to global vacuum energy beyond it. Once vacuum energy and curvature are made explicitly observer–relative, the cosmological constant puzzle is reinterpreted as a misidentification of theoretical domains rather than a physical inconsistency. 2 Motivation: Established Observer–Dependent Vacuum Effects Before introducing the formal framework, it is useful to recall that quantum field theory already contains several well–tested phenomena in which different observers disagree on the particle content or thermal character of a single global state. 1. Unruh Effect. An accelerated observer perceives the Minkowski vacuum as a thermal state with temperature T∼a, while an inertial observer detects no particles [1]. 2 2. Gibbons–Hawking Effect. A comoving observer in de Sitter spacetime measures a temperature T∼H[3]. 3. Hawking Radiation. A distant observer detects thermal radiation from a black hole, while a freely falling observer experiences no local flux [?, 2]. 4. Horizon Entropy. Entropy associated with a causal horizon depends on which observer possesses that horizon, implying an observer–dependent count of inaccessible degrees of freedom [4, ?]. 5. Algebraic QFT. Restricting a global state to a local algebra generically produces a thermal (KMS) state, even if the full state is pure [6, 7]. 6. Thermodynamic Gravity. Einstein’s equation can be derived from the Clausius relation δQ =T δS applied to local Rindler horizons [5]. Taken together, these results indicate that the vacuum structure of quantum fields is not absolute but depends on the observer’s causal access, acceleration, and horizon geometry. The present work asks whether the same principle should apply to vacuum energy itself. 3 Principle: Relativity of Reconstruction The physical world is not given as a single, observer–independent spacetime equipped with a universal stress tensor. Instead, each observer reconstructs an effective spacetime and an effective energy content from the subset of quantum degrees of freedom accessible within their causal domain. Axioms (Draft) 1. Observer = Access. An observer is defined not by consciousness or measurement, but by the subset of quantum degrees of freedom they can causally interact with. Formally, each observer qis associated with a local algebra of observables Aq. 2. Reconstruction. Each observer reconstructs an effective spacetime and vacuum energy from their accessible algebra Aq. 3. Overlap Consistency. Where two observers’ domains overlap, their reconstructions must agree on all physically communicable observables. On the intersection algebra Aq1∩q2, expectations coincide: Eq1◦Eq2=Eq2◦Eq1=Eq1∩q2. 4. No Privileged Reconstruction. No single observer’s reconstruction is ontologically preferred. Physical content resides in the relations between reconstructions, not in any one reconstruction alone. 3 Operational Principle From these axioms we postulate a single operational rule, the Relativity of Reconstruction: ⟨A(q)⟩= Tr[ρ A(q)] , A(q)∈ Aq. Different observers correspond to different operator algebras acting on the same global quantum state ρ. No collapse postulate is required. Unruh, Hawking, and Gibbons–Hawking thermality follow as special cases of this rule. 4 Observer–Relative Vacuum Energy and Curvature The Relativity of Reconstruction implies not merely that observers interpret the same global state differently, but that vacuum energy and curvature are fundamentally observer–relative reconstructions rather than universal geometric scalars. We make this ontological claim explicit. 4.1 Definition: Observer–Relative Vacuum Energy For an observer qwith associated local algebra Aqand admissible reconstruction map Fq: A→Aq, we define the physical vacuum energy not as the global quantity ρQFT, but as: ρphys vac (q)≡ρeff(q) := (FqT00)ρ.(4) This quantity depends only on the degrees of freedom accessible to qand reduces to standard vacuum energy in the limit that Fqbecomes the identity map. 4.2 Definition: Observer–Relative Stress Tensor T(q) µν ≡Fq(⟨Tµν⟩ρ), T(q) µν ∈ Aq.(5) T(q) µν is a genuine tensor within the causal patch of the observer, agrees with other observers on overlap algebras Aq1∩q2, and satisfies T(q) µν → ⟨Tµν⟩ρas H→0 (no horizon). 4.3 Observer–Relative Semiclassical Gravity We propose that curvature responds to the reconstructed, not global, stress tensor: Gµνg(q)= 8πG T(q) µν ,(6) where g(q)is the geometry reconstructed from the subalgebra Aq. Different observers reconstruct geometries that agree on overlaps but need not globally coincide. 4 4.4 Cosmological Constant as Reconstruction Parameter For de Sitter observers, ρeff(q) = ρphys vac (q)∼H2M2 Pl,(7) so the cosmological “constant” becomes: Λ(q)≡ρeff(q)∼H2M2 Pl.(8) Thus Λ is not a universal bare parameter but an emergent, observer–dependent reconstruction tied to causal access and horizon thermodynamics. 4.5 Conceptual Summary access ⇒ Aq⇒T(q) µν ⇒g(q)⇒curvature as reconstruction.(9) Vacuum energy is therefore not a fixed scalar woven into spacetime, but an observer– relative expectation on a restricted algebra. Curvature responds not to the global state but to reconstructed energy conditioned by horizon structure and information accessibility. 5 Global Quantum State, Patchwise Spacetimes The operational principle introduced above implies a distinctive ontology: the quantum state of the world may be global, yet the spacetime reconstructed from it is local and observer– dependent. This section summarizes that hybrid viewpoint and its consequences. Postulate (Global Quantum, Local Spacetime). The fundamental description is a single global quantum state ρ. Spacetime, vacuum energy, and gravitational dynamics are not intrinsic properties of ρitself, but arise from observer–dependent reconstructions based on the subset of ρaccessible within each observer’s causal domain. Interpretation. This hybrid combines a global quantum ontology with patchwise emergent spacetimes: •Quantum/boundary/fundamental level — a global state ρ, shared by all observers in principle. •Spacetime/bulk/emergent level — local reconstructions ρqand stress tensors Tµν (q) defined only within each causal patch. Different observers therefore inhabit consistent but distinct effective spacetimes. Agreement in overlap regions is enforced by the consistency axiom Eq1∩q2. Mathematical Aside. This structure mirrors the algebraic hierarchy Aq1∩q2⊂ Aqi⊂ A, with conditional expectations Eqiimplementing coarse–graining from the global algebra. At the boundary (AdS-like picture) the state ρis global; in the bulk (dS-like picture) each ρqis a reduced KMS state. 5 Conceptual Consequences. •The quantum world is global and unchanging. •Each observer reconstructs a local spacetime and vacuum energy. •Overlap consistency ensures physics agrees where observers meet. •The framework unifies insights from holography (global ρ), relational quantum mechanics (observer–relative states), and thermodynamic gravity (local horizons). Synthesis. This postulate does more than reconcile quantum mechanics with local spacetime physics—it unifies several mature but previously disconnected ideas. It may represent the missing synthesis between: •holography, where a single global state encodes many bulk reconstructions; •quantum information, where the state is primary and geometry is emergent; •general relativity, where spacetime is an observer’s causal structure; •thermodynamic gravity, where horizons define local physics; •algebraic QFT, where different observers correspond to different operator algebras. In all these languages the same pattern appears: one global state ρ−→ {many observer–dependent spacetimes}. That is not how physics is usually phrased, yet it is precisely what existing frameworks are already hinting at (holography and modular reconstruction: [17, 18, 19, 20]; relational and quantum–informational approaches: [12, 13, 14]). Conceptual Resolution. The hybrid view also dissolves a long–standing tension: quantum theory insists on a single, global state that evolves unitarily, while general relativity describes a network of local, observer–specific spacetimes. Here we retain both truths simultaneously: a global quantum state and patchwise emergent spacetimes. This is closely related to what has been called “quantum–first gravity” or “holographic reconstruction,” but presented here in a fully observer–dependent form. It occupies the intersection of these separate research programs; what they suggest separately, the present framework glues together. 6 Why de Sitter Space de Sitter (dS) spacetime—the maximally symmetric solution with positive curvature—provides the cleanest arena for the problem at hand. It is the classical model most closely approximating the observed accelerating universe and is characterized by a single scale H. Each inertial observer in dS is enclosed by a cosmological event horizon of radius H−1and experiences a thermal bath at temperature TdS =ℏH/2πkB. The presence of this temperature—and the associated horizon entropy—implies that no observer can access the full global state. 6 Mathematical Aside. The static patch of de Sitter space defines a natural von Neumann subalgebra Aq⊂ A consisting of operators with support inside the observer’s causal diamond. The global Bunch–Davies vacuum restricts to a thermal KMS state on Aqat temperature TdS. (Bisognano–Wichmann; Gibbons–Hawking.) Three features of this geometry are crucial: 1. There is no spatial boundary on which a universal stress tensor or global energy can be defined. 2. Every observer possesses a finite causal patch bounded by a horizon, ensuring that reconstruction is inherently local. 3. The only operational notion of energy is that measurable within that patch, through detector responses at scale H. Together these properties make de Sitter the natural physical setting for an observer– dependent reconstruction of vacuum energy. Positive Curvature as Physical Necessity. Formally, the algebraic mechanism behind Relativity of Reconstruction could be written in any spacetime: all that is required is a global state, localized subalgebras, a KMS condition, and limited detector bandwidth. However, only in a positively curved universe do these ingredients acquire operational meaning. In flat or Anti–de Sitter (AdS) space an observer can, in principle, recover global information; there is no thermal barrier. In de Sitter space, by contrast, the horizon permanently hides part of the global state, enforcing the relativity of reconstruction. Comparison with AdS/CFT. Mathematically, the structure resembles subregion duality in AdS/CFT. In that correspondence, a boundary subregion Rhas an associated subalgebra ARand modular Hamiltonian KR, and expectations coincide on overlaps—precisely the content of our consistency axiom. But the physical interpretation is inverted. In AdS, the boundary encodes all information: reconstruction is global. In de Sitter, each observer’s horizon enforces a strict cutoff: reconstruction is local and incomplete. Thus the same algebraic language of conditional expectations and modular flow applies, but only the positively curved case connects directly to the cosmological constant problem. Synthesis. We may summarize the contrast succinctly: AdS: global reconstruction (information recovery) ⇐⇒ dS: relative reconstruction (information inaccessibility). In this sense, Relativity of Reconstruction plays in de Sitter spacetime the role that holographic duality plays in Anti–de Sitter: it bridges a single global quantum state and the many local, observer–dependent spacetimes reconstructed from it [9, 10, 11]. Formally the mechanism is algebraic and universal, but it becomes physically significant only in the presence of positive curvature and cosmological horizons. 7 7 Reinterpreting the Vacuum Energy Problem The cosmological constant problem is usually phrased as a catastrophic mismatch between two numbers that are assumed to describe the same vacuum energy: ρQFT ∼M4 Pl, ρΛ,obs ∼H2M2 Pl. The ratio between them is the familiar factor of 10120. In the traditional view this discrepancy signals a failure of quantum field theory, general relativity, or their combination. Operational Reinterpretation. In a reconstruction-based framework these two quantities need not represent the same observable. The first is a global ultraviolet property of the state ρ, obtained by integrating over all modes of the field. The second is an infrared, observer–dependent reconstruction performed within a single causal patch of de Sitter spacetime, where an observer has access only to modes with typical frequency ω≲H. The mismatch is therefore not necessarily a physical inconsistency, but a category error: it compares a global average with a local reconstruction. Mathematical Aside. The expectation value of the stress tensor relevant to observation is not ⟨Tµν⟩ρover the entire Hilbert space, but the restricted trace ⟨Tµν⟩ρq= Tr[ρ Eq(Tµν)], where Eqis the conditional expectation onto the observer’s algebra Aq. The ultraviolet modes that dominate ρQFT lie outside Aqand contribute no observable effect. Conceptual Shift. Under Relativity of Reconstruction, vacuum energy is not a universal scalar but an observer–dependent expectation value. Different observers reconstruct different effective stress tensors Tµν (q), each defined only within their causal domain. The cosmological constant problem then dissolves: The quantities ρQFT and ρΛ,obs belong to different reconstruction domains and were never meant to be numerically identical. The paradox arises only if one assumes that all energy in the global state must be gravitationally active for all observers. Relation to QFT and GR. Importantly, this reinterpretation does not modify either quantum field theory or general relativity. It reframes the question: should spacetime curvature couple to the globally defined ⟨Tµν⟩ρ, or to the observer–restricted ⟨Tµν⟩ρq? The present framework does not answer this definitively, but provides a consistent setting in which the issue can be posed without contradiction. Summary. The “10120 problem” may therefore be not a failure of the laws of physics, but a confusion of categories. Global QFT energy and observed cosmological energy correspond to different levels of reconstruction—global versus patchwise, ultraviolet versus infrared. The Relativity of Reconstruction does not claim to solve the cosmological constant problem; it clarifies why the paradox arises and how it might disappear once the observer–dependence of vacuum energy is made explicit. In particular, ρeff(q) must be understood not as a renormalized global quantity but as the observer–relative reconstruction defined in Eq. (2). 8 8 Worked example: Gaussian RoR filter and static– patch modular term In this section we show, in a fully explicit toy model, how: a purely geometric RoR coarse–graining generically produces an effective vacuum energy density scaling as ρgeom eff ∝H4,(10) while adding a static–patch modular/thermal term introduces an additional factor of order M2 Pl H2,(11) so that the combined observer–relative stress tensor yields ρeff(q)∼H2M2 Pl .(12) We work in 4D de Sitter with Hubble scale Hand Planck mass MPl (so 1/G =M2 Pl), and consider a static–patch observer qwith causal domain Dq. 8.1 Geometric RoR operator Fgeom q We take as RoR filter a normalized, positive, radial kernel on the static patch: Kq(x, y) = KddS(x, y); ℓq, ℓq=α H,(13) where ddS is the de Sitter invariant distance in the static patch, αis a dimensionless constant, and K≥0 is a single–scale profile (Gaussian, Yukawa, Mat´ern, etc.). For any local observable Oin the global algebra A, the geometric RoR operator is (Fgeom qO)(x) := ZDq dµq(y)Kq(x, y)O(y) ZDq dµq(y)Kq(x, y) .(14) Key properties. •Positivity & normalization. Kq≥0 and the denominator is strictly positive, so Fgeom qis a normalized positive kernel operator. •Causal support. The integration domain is Dq; the filter only uses data inside the static patch. •Static–patch isometries (c–axis). Because Kqdepends only on ddS, it is invariant under static–patch isometries. 9 AQFT fact 2: kernels that differ by a change of shape but share the same support and normalization define CP maps in the same equivalence class. They correspond to different choices of “smeared test functions” in the net of algebras, but they induce the same scaling behavior on local fields (see Sec. 9.3). Thus AQFT does not distinguish between Gaussian, Mat´ern, Yukawa, bump, or exponential kernels—these are all equivalent as normal states on a local algebra. 9.2 Allowed Kernels Form a Single Universality Class The RoR construction imposes only the following structural constraints on the kernel family: •Positivity: Kq≥0 (ensures CP-ness and monotonicity on states). •Normalization: RKq= 1 (keeps Fqunital). •Radial dependence: Kq=KddS(x, y); ℓq(ensures isometry invariance and the c-axis constraint). •Single length scale ℓq: K(d;ℓq) = ℓ−3 qK0(d/ℓq),(61) (ensures holographic scaling). •Spectral decay: the Fourier transform W(kℓq) decreases smoothly for kℓq≫1 (ensures coarse-graining actually suppresses UV modes). •Locality of support: the kernel is built only from Dq. •Monotonicity: ∂pℓq(p)≥0 for the s-axis. Crucially, these constraints do not fix the functional form of K0. Any smooth, positive, radial, single-scale profile produces an admissible CP map in the same Hochschild cohomology class of local endomorphisms of A(Dq). Thus the kernel is arbitrary up to its universality class. Examples belonging to this class: •Gaussian, •Exponential/Yukawa, •Mat´ern family (all ν > 0), •Rational spectral kernels, •Compact bump functions, •Heat kernels of Laplace-type operators, •Any convex combination of the above. All generate the same AQFT properties and the same scaling dimension under dilation of the single scale ℓq. 16 9.3 Dimensional Universality of the ℓ−4Scaling In algebraic QFT on curved space, energy density is a local, dimension-4 operator. Under coarse-graining by a single-scale smearing function with width ℓq, a local operator Oof engineering dimension ∆ is mapped to an effective operator with expectation value scaling as ⟨Fgeom qO⟩∝ℓ−∆ q,(62) independently of the smearing shape. This follows from: •the microlocal spectrum condition, •the locality of the scaling limit (Haag–Narnhofer–Stein), •the equivalence of scaling nets under change of smearing functions (Buchholz–Verch). Thus for the stress-tensor energy density T00, with ∆ = 4, ρgeom eff (ℓq) = ⟨Fgeom qT00⟩∝ℓ−4 q.(63) No kernel shape can change the power −4. Changing kernels changes only the finite coefficient Cgeom, never the scaling exponent. This is the AQFT reason the “Gaussian toy model” and all other smooth kernels give the same H4result when ℓq∝1/H. 9.4 Why Kernel Choice Does Not Affect the Final H2M2 Pl Scaling The observer-relative coarse-graining map is Fq=λqFgeom q+(1−λq)Fmod q.(64) The geometric piece always contributes a term of order H4. The modular (gravitational) piece always contributes a term of order H2M2 Pl. The ratio of these contributions is universally ρmod ρgeom ∼M2 Pl H2Cgeom ≫1 for any Cgeom =O(1) .(65) Thus: •Different kernels give different Cgeom, •but all are O(1), •and the huge factor (MPl/H)2always ensures the modular term dominates. Hence the prediction ρeff(q)∼H2M2 Pl (66) is stable under all allowed kernel choices. 17 9.5 Summary From the AQFT viewpoint: •All admissible kernels correspond to the same class of completely positive, unital, normal maps on A(Dq). •The kernel’s detailed shape does not enter the scaling limit or its algebraic properties. •Dimensional analysis enforced by the scaling-net framework ensures Fgeom qcontributes ∼ℓ−4. •The gravitational modular term contributes ∼H2M2 Pl. •Their competition is universally dominated by the modular term. Therefore, the kernel family is arbitrary up to mild structural constraints, and the final vacuum energy scaling is independent of that choice. 10 Open Questions and Research Directions The Relativity of Reconstruction is not a mere interpretive shift; it opens a concrete research program. Each component of the framework corresponds to a technical problem that can be precisely formulated and, in principle, solved. 10.1 de Sitter Reconstruction and Energy The Bunch–Davies vacuum is de Sitter invariant yet admits no global timelike Killing vector. Each static observer measures a thermal response at TdS =ℏH/2πkB, yielding an operational energy density ρobs ∼H4. To connect this to the gravitationally inferred ρobs Λ∼H2M2 Pl, one must link reconstructed energy to curvature through horizon entropy and temperature. Mathematical Aside. When the modular Hamiltonian is local, Kq=−log ρq= RΣζνTµνdΣµ,the entanglement first law δS =δ⟨Kq⟩relates horizon–entropy change to reconstructed energy [17, 18]. This relation suggests that the observed ρΛ,obs ∝H2M2 Pl may arise naturally from the product of horizon entropy (∼M2 Pl/H2) and thermal energy scale (∼H4), without any fine tuning. 10.2 Mathematical Target: Covariant Filtered Stress Tensor The next step is to identify what quantity Einstein’s curvature should couple to in a reconstructionbased framework. A key objective is to replace the coupling to a global stress tensor by an observer–restricted (reconstructed) tensor, Gµν = 8πG Tµν (q), Tµν (q)=Eq(⟨Tµν⟩), where Eqacts as a causal coarse–graining map corresponding to the observer’s horizon or acceleration. This tensor should satisfy: 18 Covariance. Tµν (q)is a genuine tensor; observer–dependence enters only through causal structure, not coordinates. Conservation. Either strong conservation, ∇µTµν (q)= 0, holds within the patch, or a controlled modification with flux term ∇µTµν (q)=Jν (q)consistent with the Bianchi identities. Correct de Sitter scaling. Applied to vacuum in de Sitter, the construction should yield Tµµ(q)∼H2M2 Pl, suggestively captured by the “entropy ×temperature” estimate. Mathematical Aside. Smooth smearing of Tµν with test functions supported within the causal patch (Fewster; Ford–Roman) offers a rigorous model of the filter. Quasilocal formulations such as Brown–York energy, canonical–energy constructions, and stochastic or coarse–grained gravity approaches [15, 16, 17, 20] may provide covariant realizations of the observer–restricted stress tensor Tµν (q). Formally, Tµν (q)can be regarded as an operator–valued map Tµν (q)=F(q)[⟨Tµν⟩], where F(q)reduces to the identity when the observer has full access (no horizon) and otherwise acts as a causal–patch projection or modular restriction preserving conservation. 10.3 Concrete Open Problems 1. Filtered Stress Tensor. Construct a covariant, observer–restricted Tµν (q)that is conserved within the causal patch. 2. Einstein Equation with Reconstruction. Determine whether curvature should couple to global or reconstructed energy, and identify what replaces ∇µTµν = 0 if only patchwise conservation holds. 3. Explicit de Sitter Implementation. Combine Gibbons–Hawking temperature and horizon entropy with algebraic restriction to compute the reconstructed vacuum energy in dS. 4. Transitions Between Observers. Formalize update rules when observers with different horizons exchange information, ensuring consistency on overlap regions. 5. Holographic Version. Reformulate Relativity of Reconstruction as bulk emergence from subregion density matrices, analogous to entanglement–wedge reconstruction in AdS/CFT but without requiring a global boundary. 6. Operational Definition of Λ.Clarify whether the cosmological constant is a bare coupling in the action or an emergent parameter tied to horizon thermodynamics. 7. Empirical Probes. Identify analogue systems (accelerated detectors, Rindler cavities, trapped ions) that could display similar reconstruction filtering. 19 Synthesis. These problems collectively define the mathematical roadmap implied by Relativity of Reconstruction. They show how the framework can be made fully quantitative and potentially predictive. In particular, constructing a covariant filtered stress tensor would supply the missing link between operationally defined vacuum energy and gravitational dynamics—the step required to turn the conceptual consistency of this approach into a complete theory. 11 Relation to Type II Horizon Algebras, QRFs, and Observer–Dependent Entropy Recent advances in gravitational algebra, modular theory, and quantum reference frames have dramatically clarified the operator-algebraic structure of horizon-adjacent degrees of freedom. A sequence of papers culminating in the works of Chandrasekaran, Longo, Penington, and Witten (CLPW) [23], and subsequently extended by Kudler-Flam, Leutheusser, Satishchandran and collaborators [24], showed that once gravitational dressing is properly accounted for, the would-be Type III local algebra of a black hole or de Sitter wedge is promoted to a Type II crossed product. The existence of a semifinite trace on this algebra makes it possible to interpret generalized entropy as an honest von Neumann entropy. This result represents a conceptual leap: gravity reshapes the operator algebra in such a way that a meaningful notion of horizon entropy emerges directly from the algebraic structure. The subsequent mathematical generalizations by Fewster, Janssen, Loveridge, Rejzner, and Waldron [25] provided the first complete operator-algebraic derivation of these Type II phenomena from relativistic measurement theory and quantum reference frames (QRFs). In this framework, the physical algebra of a region is obtained as a crossed product of the field algebra with the symmetry group of the QRF, and the existence of a trace (and thus of a Type II1factor) follows from KMS and modular conditions obeyed jointly by the frame and the field. These results place the CLPW construction on a rigorous algebraic footing: the gravitational subsystem is not an add-on, but an intrinsic part of the local observable algebra. A parallel line of work by De Vuyst, Eccles, H¨ohn, and Kirklin [26] then used the QRF framework to show that gravitational entropy is in general observer-dependent. Different QRFs correspond to different crossed products, yielding distinct Type II factors and hence different von Neumann entropies. Their companion conceptual analysis [27] makes explicit that this dependence is not a peculiarity of the constructions: it is a structural feature of any situation in which the “observer” participates in the definition of the algebra of physical observables. Together, these developments demonstrate a profound but limited principle: The gravitationally dressed algebra of observables associated to a horizon admits a trace, and the resulting generalized entropy is a QRF-dependent von Neumann entropy arising from a Type II crossed product. This is already a major departure from traditional treatments of gravitational entropy, but it focuses exclusively on the entropic sector. The reason is structural. All of the analyses cited above concern only the modular Hamiltonian, modular flow, and the entropy functional— quantities that live naturally in the domain of Tomita–Takesaki theory, and which depend 20 only on the algebra and its trace. They do not extend to general local observables such as Tµν, because the Type II crossed-product machinery provides no canonical notion of coarsegrained local operator expectation values. The trace allows one to compute an entropy; it does not provide an observer-dependent stress tensor suitable for coupling to semiclassical gravity. Indeed, as emphasized in [25], the crossed-product construction reorganizes the algebra but does not supply a local resolution scale, a local averaging prescription, or any mechanism for observer-dependent renormalization of composite operators. The present framework—the Relativity of Reconstruction (RoR)—is designed precisely to extend beyond this limitation. RoR begins from the same structural insight as the Type II program: the observer is not external to the algebra, but participates in its definition. However, instead of restricting attention to modular flow and entropy, RoR builds a family of observer-dependent coarsegraining maps Fqacting on the entire stress tensor. This introduces two elements absent from the Type II literature: 1. a local resolution scale ℓq∼H−1compatible with static-patch symmetry, enabling observer-relative renormalization of composite operators; 2. a thermodynamically fixed modular contribution that reproduces the horizon-induced H2M2 Pl term. Because the Type II crossed-product results apply only to the modular Hamiltonian and the entropy functional, they supply no mechanism for constructing an observer-dependent stress tensor and hence no way to formulate an observer-relative semiclassical Einstein equation. RoR does this by enlarging the mathematical setting: instead of relying solely on the presence of a trace, we work within the space of coarse-graining operators compatible with staticpatch symmetry, positivity, modular thermality, and the flat-space limit. This space forms a constrained manifold Mholo, and the observer–dependent effective stress tensor Tµν(q) = ⟨FqTµν⟩depends only on the coordinates of Fqwithin this manifold. Thus RoR represents the natural continuation of the Type II/QRF program: •the Type II results show that entropy is observer-dependent because the algebra itself depends on the observer; •RoR shows that the stress tensor is observer-dependent for the same reason, once one introduces the physically required coarse-graining scale and modular contribution; •the Type II works stop at entropy because the trace structure alone is insufficient to define observer-relative local operators; •RoR introduces additional geometric and thermodynamic data (the (s, u, c) coordinates) that allow observer-relative coarse-graining to be extended to the stress tensor, preserving covariance and conservation. In this way, RoR does not compete with or reinterpret the Type II horizon-algebra results: it synthesizes them. It takes the now-established lesson that “entropy is observer-dependent because the algebra is observer-dependent,” and carries it to its natural gravitational endpoint: 21 If entropy is observer-dependent because the algebra is, then any reconstructed stress tensor must likewise be observer-dependent. RoR provides the missing operator structure that allows this dependence to be computed, rendered covariant, and coupled to gravity. This logical continuation lies beyond the reach of the existing Type II techniques, not because they are incomplete, but because they are intentionally focused on modular quantities. RoR complements them by supplying the machinery needed to construct observer-relative stress tensors, thereby enabling an observer-relative semiclassical Einstein equation and resolving the radiative-stability issues that plague global treatments of vacuum energy. 12 Discussion and Outlook The Relativity of Reconstruction reframes one of the oldest puzzles in fundamental physics— the cosmological constant problem—as a question of operational domains rather than fine tuning. Vacuum energy becomes an observer–dependent reconstruction of a global quantum state, not an absolute property of spacetime. •Quantum field theory and general relativity remain intact; only the assumption of universal vacuum accessibility is revised. •The ultraviolet contribution to the vacuum state remains present in ρ, but is operationally inaccessible to any observer with finite acceleration or horizon. •The key technical challenge is to construct a covariant, observer–restricted stress tensor Tµν (q)consistent with the Bianchi identities and the observed de Sitter scaling ρobs Λ∼ H2M2 Pl. •Laboratory analogues—accelerated detectors [21, 22], Rindler cavities, and quantum simulation platforms—may eventually probe aspects of observer–dependent reconstruction experimentally. The broader implication is that spacetime and energy are not absolute features of the world but emergent, relational reconstructions from a single underlying quantum state. The Relativity of Reconstruction does not modify quantum field theory or general relativity. It removes only the assumption that the vacuum energy entering the Einstein equation must be a universal, observer–independent scalar. Whether this conceptual shift ultimately resolves the cosmological constant problem will depend on developing the filtered stress tensor in a fully covariant and conserved form—a task left for future work. 13 Kernel-Operator Reformulation of RoR Filters The Relativity-of-Resolution (RoR) framework assigns to each static-patch observer qa coarse-graining operator Fq:O(Dq)→ O(Dq), 22 acting on local observables restricted to the observer’s static patch Dq⊂dS4. In this section we reformulate these operators in explicit kernel-operator language and define a mathematically well-posed class HdS of admissible RoR filters. This replaces earlier heuristic definitions with a controlled, representation-independent operator family possessing positivity, normalization, covariance, and a flat-space limit. 13.1 Static-Patch Geometry and Notation Let (dS4, g) denote four-dimensional de Sitter spacetime with Hubble parameter H. Each observer qis associated with a static patch Dqand a natural invariant distance function ddS(x, y)onDq. We denote by µqthe static-patch measure induced by g. 13.2 The Geometric Kernel Class CdS We begin by defining the class of covariant, normalized, positive integral kernels that implement the geometric (UV-blurring) part of the RoR coarse-graining. [de Sitter kernel class] Let CdS denote the set of all measurable kernels KH:Dq×Dq→ R≥0satisfying: 1. Positivity: KH(x, y)≥0. 2. Normalization (unitality): ZDq KH(x, y)dµq(y) = 1 for all x∈Dq. 3. de Sitter covariance: KH(x, y) = KH(ddS(x, y)), for some scalar profile KHdepending only on invariant distance. 4. Single-scale structure: KHdepends on a single geometric length scale ℓ(H)∼H−1. That is, KH(r)=ℓ(H)−3Kr ℓ(H), with Ka fixed dimensionless function. 5. Identity limit: lim H→0KH(x, y) = δ(x−y) in the sense of distributions on compact subsets. We call CdS the geometric kernel class. 23 In applications to the energy density T00 we will restrict to equal–time spatial slices inside Dq. In that case the kernel acts on a three–dimensional spatial hypersurface, so the natural normalization scale is ℓ(H)−3. The ℓ−4scaling of the coarse–grained energy density then arises from the dimension (4) of T00 and the single–scale structure, not from the prefactor in the kernel definition. This class abstracts the essential features of isotropic, resolution-limited coarse-graining consistent with static-patch symmetry. 13.3 Geometric Operators Every kernel KH∈CdS defines an operator acting on local observables. [Geometric RoR operator] Given KH∈CdS, define Fgeom qO(x) := ZDq KH(x, y)O(y)dµq(y), for any local observable O:Dq→R. By construction, these operators implement a positive, normalized smoothing of Oat the de Sitter scale ℓ(H). 13.4 Modular Contribution from Horizon Thermodynamics In addition to geometric smoothing, RoR includes an observer-dependent modular contribution associated with the static-patch thermal state. The modular Hamiltonian of the Bunch–Davies vacuum restricted to Dqinduces a universal, patch-constant stress-tensor term of the form Tmod µν (H):=−α(H)M2 PlH2gµν, where α(H) is a dimensionless function with α(0) = 0 ensuring the flat-space limit. This term encodes the horizon-thermodynamic contribution to the observer-relative energy density and is independent of the UV details of the observable. 13.5 The Full RoR Filter Class HdS We now combine the geometric and modular pieces. [Admissible RoR filters] Let HdS denote the class of operators Fqacting on local observables Oin Dqby Fq[O]:=λqFgeom q[O] + (1 −λq)Omod, where: •0≤λq(H)≤1 is a mixing parameter constrained by symmetry and the identity limit λq(H)→1 as H→0; •Fgeom qarises from any KH∈CdS; •Omod denotes the modular contribution, which for the stress tensor is Tmod µν (H) as above. 24 Every Fq∈HdS is called an admissible RoR filter. Thus HdS is a constrained manifold of operators labeled by the smoothing scale ℓ(H), modular weight λq(H), and choices within the geometric kernel class CdS. 13.6 Basic Properties Every Fq∈HdS satisfies: 1. Positivity: If O(x)≥0 then (FqO)(x)≥0. 2. Normalization: Fqpreserves constant observables. 3. Locality in the static patch: supp(FqO)⊆Dq. 4. Static-patch covariance: For any isometry ϕpreserving Dq, Fq[O◦ϕ] = (FqO)◦ϕ. 5. Flat-space limit: As H→0, Fq→id in the sense of distributions. 6. Two-channel structure for stress tensors: Fq[Tµν]=λqFgeom q[Tµν]−(1 −λq)α(H)M2 PlH2gµν. [Sketch] Positivity and normalization follow from the kernel axioms and the convex form of Definition 13.5. Covariance follows from the invariance of ddS and µq. The identity limit follows from Definition 13.2(5). Locality is immediate from the domain of integration. The stress-tensor decomposition is algebraic. This kernel-operator formulation provides a mathematically transparent foundation for all subsequent RoR constructions, including the derivation of the observer-relative vacuum energy density. 13.7 Characterization of the Admissible RoR Operator Class The preceding definitions specify the geometric kernel class CdS and the full RoR class HdS. To complete the construction we now show that HdS can be characterized as a minimal class of operators acting on static-patch observables that satisfies the following physically mandated conditions: 1. positivity on observables, 2. normalization (constants preserved), 3. static-patch covariance, 4. single-scale ultraviolet blurring at the de Sitter length ℓ(H), 5. a thermal modular contribution fixed by the static-patch KMS state, 25 of a UV–completed quantum field theory whose short–distance behavior is that of a free relativistic field. Let Fq∈HdS be any RoR filter as defined above, associated with the static patch Dqof an inertial observer q. Then the observer–relative effective stress tensor T(q) µν := Fq[Tµν] (97) has vacuum expectation value of the form T(q) µν ρ=−ρeff(H)gµν,(98) with ρeff(H)=c1H4+c2H2M2 Pl,(99) where: 1. c1is a dimensionless constant of order unity determined by the UV theory and the kernel profile K0(but not its scale), and is independent of the detailed shape of K0 within the single–scale class Cgeom dS . 2. c2is a dimensionless constant of order unity determined by the horizon thermodynamics (through TdS,SdS) and the choice of static–patch volume convention, and is independent of the kernel shape. 3. In the flat–space limit H→0 one has lim H→0ρeff(H)=0,lim H→0T(q) µν ρ= 0,(100) so the RoR effective stress tensor reduces to the usual Minkowski vacuum stress tensor. [Proof (sketch)] We split the proof into geometric and modular contributions. (1) Geometric contribution. For the geometric part we consider Tgeom µν (x) := (Fgeom qTµν)(x) = ZDq KH(x, y)Tµν(y)dµq(y).(101) By construction, KH∈Cgeom dS is a positive, normalized, single–scale kernel with ℓ(H) the only length scale. In a locally flat coordinate patch near the observer, the energy density T00(x) is a dimension–4 local operator. The kernel–universality proposition for such operators implies (Fgeom qT00)(x)ρ=CKℓ(H)−4,(102) with CK=O(1) independent of the detailed kernel shape K0. For ℓ(H)∝1/H at late times this gives (Fgeom qT00)(x)ρ=c1H4,(103) with c1=O(1) absorbing numerical factors and the UV renormalization chosen for the vacuum. Static–patch covariance and de Sitter invariance of the Bunch–Davies state then imply that Tgeom µν (x)ρ=−c1H4gµν(x),(104) 32 i.e. the geometric contribution is homogeneous, isotropic, and of perfect–fluid form with effective density ∝H4. (2) Modular contribution. By Proposition 15.1, any horizon–induced energy density scale built from the static–patch thermodynamics must be of the form ρmod(H)=cmod H2M2 Pl,(105) with cmod =O(1). Modeling the modular term as a homogeneous, isotropic perfect fluid Tmod µν (H)=−ρmod(H)gµν =−cmod H2M2 Pl gµν,(106) we obtain its vacuum expectation value directly: Tmod µν (H)ρ=−cmod H2M2 Pl gµν.(107) (3) Full RoR filter. The full RoR filter combines the two contributions as (FqTµν)(x) = λq(H)Tgeom µν (x) + 1−λq(H)Tmod µν (H).(108) Taking the vacuum expectation value and using linearity, T(q) µν (x)ρ=λq(H)Tgeom µν (x)ρ+1−λq(H)Tmod µν (H)ρ.(109) Substituting the expressions above gives T(q) µν (x)ρ=−λq(H)c1H4+1−λq(H)cmod H2M2 Plgµν(x).(110) Defining c2:= 1−λq(H)cmod,(111) and noting that for late–time cosmology H≪MPl one has H2M2 Pl ≫H4, we can write ρeff(H):=λq(H)c1H4+1−λq(H)cmod H2M2 Pl =c1H4+c2H2M2 Pl,(112) with c1, c2=O(1) encapsulating the dependence on λq(H), K0, and the thermodynamic conventions. The flat–space limit conditions λq(H)→1 and ρmod(H)→0 as H→0 ensure that ρeff(H)→0 and hence ⟨T(q) µν ⟩ρ→0 in the Minkowski limit. The theorem shows that, within the RoR framework and the operator class HdS, the effective vacuum energy reconstructed by any static–patch observer has a universal two– term structure ρeff(H) = c1H4 |{z} geometric / UV +c2H2M2 Pl | {z } modular / horizon .(113) The H4term is the familiar QFT contribution coarse–grained by a single–scale kernel, while the H2M2 Pl term is forced by horizon thermodynamics. Neither term is the result of fine– tuning; both follow from structural inputs (kernel scaling and de Sitter thermodynamics) that are robust under changes of kernel shape and under addition of further UV degrees of freedom, up to order–one coefficients. 33 16 The Holographic Manifold of Observer–Relative Filters In the RoR framework, the map Fq:A(Dq)→ A(Dq) is not an arbitrary coarse–graining operator. Rather, the admissible observer–relative filters form a constrained, low–dimensional manifold of operators. This subsection describes that manifold and the natural coordinates on it. 16.1 Geometry of the Operator Space Let HdS denote the class of physically admissible RoR filters on the static patch Dqof an inertial observer qin four–dimensional de Sitter spacetime. By definition, an operator Fq∈HdS must satisfy: •positivity, •normalization (unitality), •locality on the static patch, •static–patch isometry covariance, •a single geometric scale ℓ(H) with ℓ(H)∼α/H, •a de Sitter–consistent modular component fixed by horizon thermodynamics, •a good flat–space limit H→0. These requirements drastically restrict the possible maps Fq. Once kernel shape is modded out (by universality of leading scaling), the remaining freedom collapses to a finite– dimensional family of operator choices. We call this the holographic manifold of observer filters: Mholo ⊂HdS. 16.2 Coordinates on the Manifold A convenient parametrization of Mholo is given by three geometric axes: (s, u, c)∈R3, corresponding to scale, quotient mixing, and symmetry. 34 1. The scale coordinate s.The coordinate slabels the geometric coarse–graining scale. Concretely: s= log1 ℓ(H),with ℓ(H)∼α H. Movement along the s–axis corresponds to changing how much the observer resolves within their static patch. Physical constraints fix s(H) nearly uniquely: s(H) = log(H/α). Thus sis not a free knob: it is a coordinate describing the geometric location of Fqon the manifold. 2. The mixing coordinate u.The modular/geometric mixing is encoded in a coordinate u, which can be taken as u≡λq(H)∈[0,1], with λq(H) the weight assigned to the geometric kernel component. The orthogonal complement 1 −λq(H) controls the modular (horizon thermodynamic) component. The key point is: λq(H)is fixed by physical constraints, not freely chosen. Specifically, the following requirements restrict u(H): flat limit: lim H→0λq(H)=1, thermodynamic consistency: ρmod(H)=c H2M2 Pl, positivity and covariance: λq(H)∈[0,1]. Thus userves as a coordinate on Mholo, not an arbitrary degree of freedom introduced by hand. 3. The symmetry coordinate c.The coordinate clabels which subgroup of the de Sitter isometry group is preserved by the operator. For an observer in a static patch, symmetry imposes: c= 0 ↔static–patch isometries preserved. Departures from c= 0 correspond to breaking static–patch symmetry or introducing anisotropic coarse–graining. Such operators lie outside the admissible class HdS; hence in practice, cis fixed. 16.3 Structure of the Manifold The admissible filters take the form Fq(s, u;c) = u Fgeom q(s) + (1 −u)Fmod q, with: 35 •Fgeom q(s) determined by a single geometric scale ℓ(H), •Fmod qdetermined uniquely (up to O(1) factors) by de Sitter thermodynamics, •u=λq(H) lying on a physically determined curve u(H)⊂[0,1]. Thus the space of all possible filters collapses to a trajectory H7−→ (s(H), u(H), c = 0) ∈ Mholo ⊂HdS, illustrating the central point: RoR filters are not arbitrary constructions. They are points on a holographically constrained manifold of physically allowable observer maps. The functions λq(H)and γ(H)(or their analogues) are coordinates on this manifold, not tunable parameters. 16.4 Physical Interpretation The manifold structure reflects two deep principles: 1. Local resolution (scale): The observer can only resolve structure down to ℓ(H), set by horizon size. 2. Thermodynamic dressing (quotient): The observer must incorporate the universal modular energy associated with the static–patch horizon. 3. Symmetry (isometries): Observer filters must respect the static–patch symmetry subgroup. These three requirements fix the class of allowed operators. Thus the “freedom” in defining Fqis greatly overstated if the operator is treated as arbitrary. Once physical consistency is enforced, Fqoccupies a sharply restricted holographic submanifold of operator space. 16.5 Physics Interpretation of the (s, u, c)Coordinates The manifold Mholo of admissible observer filters Fq∈HdS can be parametrized abstractly by a set of coordinates (s, u, c), following the structural analysis of holographic operators in [?]. In the present framework these coordinates acquire direct physical meaning. This subsection provides the precise translation from the abstract operator-theoretic parametrization to physically interpretable degrees of freedom associated with static-patch coarse-graining in de Sitter spacetime. 36 16.5.1 Summary of the operator coordinates In the purely operator-theoretic description, the coordinates (s, u, c) play the following roles: 1. s: determines the resolution scale of the kernel component, 2. u: determines the relative modular–geometric mixing (a quotient-like direction already familiar in the analysis of normalized attention kernels), 3. c: labels directions corresponding to internal symmetries or isometries acting on the kernel without changing its algebraic structure. In the RoR setting these acquire clear geometric and thermodynamic meaning. 16.5.2 The s–coordinate: geometric resolution scale Let KH∈CdS denote a normalized, positive, static-patch covariant kernel with a single characteristic length scale l(H). The coordinate sis defined abstractly as the ratio s=lq H−1=Hlq,(114) where lqis the physical smearing radius experienced by observer q. Static-patch covariance and the requirement of a good flat-space limit imply that lqmust scale as1 lq(H) = α H−1,(115) with α=O(1). Thus sis not a tunable parameter of the theory; rather, s=α, (116) a constant fixed by the admissible kernel-space CdS and by the fact that an observer’s causal domain is itself geometrically bounded by the horizon scale H−1. Physical interpretation. The s–axis encodes the only available coarse-graining scale permitted by de Sitter geometry. It is physically determined by: •the size of the static patch, •the requirement of positivity and normalization, •the de Sitter isometry group SO(1,4), •the H→0 limit, which forces lq→ ∞. Thus, although sappears as a “degree of freedom” in the abstract manifold, it is frozen by geometric constraints in the physical theory. 1See the kernel-operator definition and Proposition ??. 37 16.5.3 The u–coordinate: geometric–modular mixing In operator form, the RoR filter can be written as Fq=λqFgeom q+ (1 −λq)Fmod(H),(117) where λq∈[0,1] and Fmod(H) is the modular contribution determined by horizon thermodynamics. We identify the u–coordinate with λq: u≡λq.(118) Physical constraints on u.The mixing coefficient u=λqis restricted by the following: 1. Positivity and normalization: both Fgeom qand Fmod are positive operators with unit normalization; convexity restricts u∈[0,1]. 2. Static-patch KMS condition: the modular contribution must reproduce the correct thermal behavior at the de Sitter temperature T=H/(2π). 3. Flat limit: as H→0 the modular term disappears, requiring λq(H)→1.(119) 4. Covariance: Fqmust transform appropriately under SO(1,4) isometries; this prohibits arbitrary q–dependence of λq, allowing only those forms compatible with patch symmetries. Interpretation. The coordinate uis therefore a physically meaningful, but constrained degree of freedom describing how the observer’s filter interpolates between: •a purely geometric coarse-graining with UV cutoff l−1, and •a purely modular coarse-graining determined by horizon entropy, generating the H2M2 Pl contribution. It is not a free parameter introduced by hand; it is a coordinate labeling points in the constrained operator space HdS. 16.5.4 The c–coordinates: static-patch isometries and frame choice The remaining coordinates clabel directions obtained by acting on Fgeom qwith isometries of the static patch: Fq7→ U(g)FqU(g)†, g ∈SO(1,4).(120) Because of static-patch isotropy, such transformations leave all expectation values of rotationally invariant quantities (e.g. the local energy density) unchanged. Thus the c–directions correspond to: 38 •choice of isometry frame, •orientation of coordinates on the patch, •gauge-like redundancies associated with SO(1,4). Therefore: ρeff(H)=⟨(FqT00)⟩is independent of c. (121) Interpretation. The c–axis consists entirely of unphysical redundancies: different representatives of the same equivalence class of filters. The physically distinct information resides only in the (s, u) subspace. 16.5.5 Dimensionality of the physical manifold From the analysis above: •the s–coordinate is fixed by geometry, •the c–coordinates correspond to isometries and carry no physical content for isotropic observables, •the only nontrivial continuous degree of freedom is the geometric–modular mixing u=λq. Thus the physically relevant parameter space is effectively one-dimensional: Mphys ∼ ={u∈[0,1] }.(122) Nevertheless, the embedding of this line into the larger manifold Mholo is what enforces the strict form of ρeff(H): ρeff(H)=c1H4+c2H2M2 Pl, independently of the UV field content or operator details. This follows from: 1. kernel universality for the geometric direction (H4), 2. thermodynamic uniqueness of the modular direction (H2M2 Pl), 3. closure of Mholo under addition of UV modes (radiative stability). This completes the physics translation of the (s, u, c) coordinates. 39 16.6 Radiative Stability from the Holographic Manifold We now formalize the sense in which the observer–dependent filter Fqis radiatively stable. The key point is that the admissible filters form a low–dimensional constrained manifold Mholo ⊂HdS, and renormalization—including the addition of new UV modes—acts only by moving Fqalong this manifold, never off of it. Consequently, the functional form of the effective coarse–grained energy density is fixed. [Radiative Stability on the Holographic Manifold] Let Mholo ⊂HdS denote the class of normalized, positive, static–patch–covariant operator filters with: 1. a single geometric coarse–graining scale l(H)∼H−1, 2. an H→0 flat–space identity limit, 3. a modular contribution determined by the static–patch KMS condition and horizon thermodynamics. For any such Fq∈ Mholo and any local observable Oof mass-dimension four (in particular T00), the observer–relative energy density ρeff(H)≡ ⟨(FqT00)⟩ρBD (123) takes the universal form ρeff(H) = c1H4+c2H2M2 Pl, c1, c2=O(1),(124) where: 1. the coefficient c1arises entirely from the geometric smearing by kernels in CdS, whose universality implies dimensionally that ⟨Fgeom qT00⟩ ∝ l(H)−4∼H4; 2. the coefficient c2arises from the modular (horizon) term enforced by the thermodynamic uniqueness of the static–patch KMS state and is fixed up to O(1) factors. Moreover, for any enlargement of the UV theory by additional fields, modes, or interactions, the renormalized filter Fren qremains in Mholo. In particular, adding new UV structure cannot modify the dependence of ρeff(H) on H. [Sketch of proof.] The constrained structure of Mholo implies: (1) Geometric contribution. By Proposition 14.2, any kernel in CdS with single scale l(H) acts as a smooth ultraviolet cutoff |k|≲l(H)−1on dimension-four operators. This yields ⟨Fgeom qT00⟩ ∼ l(H)−4∼H4.No kernel-shape dependence survives beyond an O(1) constant. (2) Modular contribution. Static–patch KMS symmetry and the de Sitter first law fix the renormalized modular stress tensor to have the unique de Sitter–covariant form Tmod µν (H) = −α H2M2 Pl gµν,with α=O(1) determined by the horizon entropy gradient and the Clausius relation. (3) Radiative robustness. Renormalization of Tµν or addition of new fields only introduces additional UV structure inside the convolution integrals defining Fq. However, the scale separation l(H) is fixed by macroscopic geometry, and the modular term is fixed by thermodynamics; hence renormalization acts only by adjusting coordinates along Mholo, not by departing from it. Thus the functional form H4+H2M2 Pl is stable. This result should be understood as a heuristic structural claim about how renormalization acts within Mholo, rather than as a fully rigorous renormalization–group theorem. 40 16.7 Determined vs Undetermined Structure in the RoR Operator Class The operator-class construction above fixes several aspects of the effective vacuum energy, while leaving others to be determined by additional physical input beyond the present local analysis. •Fixed by structure: –The existence of a geometric contribution to the vacuum energy with scaling ∝H4. –The existence of a modular (horizon-induced) contribution with scaling ∝H2M2 Pl. –The single-scale behavior ℓ(H)∼α/H of admissible kernels, and the associated spatial normalization. –Positivity, locality on the static patch, static-patch covariance, and the H→0 flat-space identity limit of Fq. •Left undetermined at this stage: –The detailed profile K0of the geometric kernel within its universality class (affecting CKbut not the scaling). –The order-one coefficient αin ℓ(H) = α/H. –The mixing function λq(H) beyond its limiting behavior λq(H)→1 as H→0. –The numerical coefficients c1and c2in ρeff (H)=c1H4+c2H2M2 Pl. These undetermined quantities are analogous to scheme-dependent data in renormalization: they are constrained by symmetry and locality but not fixed by them. Their precise values, or functional dependence on H, require additional dynamical or geometric input (for example, conservation conditions, matching to a global semiclassical Einstein equation, or more microscopic constructions), which lie beyond the scope of the present operator-level analysis. 16.8 Relation to No-Go Theorems and the Operator Manifold Classical no-go theorems for vacuum energy (e.g. Weinberg’s theorem) assume that the gravitational field equations couple to a single, observer-independent renormalized expectation value Tvac µν =⟨Tµν⟩ρ,(125) defined globally and invariant under all Poincar´e (or de Sitter) symmetries. Under this assumption, radiative corrections generically shift Tvac µν by O(M4 UV). In the RoR framework this hypothesis is replaced by a strictly weaker—and physically motivated—one: the gravitational field couples not to a single global vacuum tensor, but to the family of observer–relative coarse–grained tensors Tµν(q)≡ ⟨(FqTµν)⟩, q ∈static patches.(126) The admissible maps Fqlie on a constrained manifold Mholo, whose geometry is fixed by: 41 and the static–patch volume scales as Vq∼cVH−3, cV=O(1).(149) Then the only possible energy density scale that can be formed from (TdS, SdS, Vq) is ρmod(H)=Cmod H2M2 Pl, Cmod =O(1).(150) Equivalently, the “modular plate” contribution to the observer–relative stress tensor is ⟨Tmod µν (x)⟩=−ρmod(H)gµν(x)=−Cmod H2M2 Pl gµν(x).(151) Proof (thermodynamic). The static patch possesses a horizon of area A= 4π/H2with Gibbons–Hawking entropy and temperature SdS =A 4G=πM2 Pl H2, TdS =H 2π.(152) The natural horizon energy scale built from (T, S) is Ehor ∝TdS SdS =H 2π·πM2 Pl H2=1 2 M2 Pl H.(153) The static patch has proper volume Vq∼cVH−3,(154) so the associated energy density is ρmod(H) = Ehor Vq ∝M2 Pl/H H−3=H2M2 Pl.(155) All ambiguities in the definition of the volume or normalization coefficients merely shift Cmod by an O(1) factor. Thus the scale H2M2 Pl is uniquely determined by thermodynamic and semiclassical considerations. C.2 Covariance and Conservation Because the modular contribution is proportional to gµν, ∇µ⟨Tmod µν ⟩= 0 (156) holds identically. No further conditions or field equations are required. C.3 Flat–Space Limit As H→0, TdS →0, SdS → ∞, TdSSdS ∼M2 Pl H→ ∞,(157) but the static–patch volume diverges faster, Vq∝H−3→ ∞. Their ratio therefore vanishes: ρmod(H)−→ 0 (H→0),(158) consistent with the flat–space identity limit Fq→id. 48 C.4 Separation of Roles: QFT vs. Thermodynamics The geometric H4term arises from coarse–graining the ultraviolet structure of the BD vacuum in QFT. The horizon–induced H2M2 Pl term arises instead from semiclassical gravitational thermodynamics and is not obtainable from any free–field calculation. The observer–relative effective stress tensor therefore has two conceptually distinct contributions: ⟨T(q) µν ⟩=−CgeomH4gµν −CmodH2M2 Plgµν,(159) with the second term dominating for H≪MPl. C.5 Summary Thermodynamic arguments uniquely fix the modular contribution to the observer–relative vacuum energy to scale as H2M2 Pl. Combined with the geometric H4term, this yields the full semiclassical observer–relative scaling used throughout the Relativity of Reconstruction framework. C.6 Combined Result: The Observer–Relative Effective Stress Tensor The geometric coarse–graining theorem (Thm. B.1) and the horizon/modular thermodynamic theorem (Thm. C.1) together determine the full observer–relative effective stress tensor appearing in the Relativity of Reconstruction (RoR) framework. Let qbe any static–patch observer in four–dimensional de Sitter spacetime. The observer– relative map Fqacts on the local stress tensor by T(q) µν ≡(FqTµν) = Fgeom qTµν +Tmod µν ,(160) where Fgeom qcoarse–grains ultraviolet degrees of freedom using a kernel in CdS, and Tmod µν is the homogeneous “modular plate” fixed by static–patch thermodynamic data. Theorem (Combined observer–relative vacuum energy). For the Bunch–Davies vacuum, ⟨T(q) µν (x)⟩BD =−Cgeom H4+Cmod H2M2 Pl gµν(x), Cgeom, Cmod =O(1).(161) Proof. By Theorem B.1, the geometric coarse–graining of the renormalized stress tensor yields ⟨Fgeom qTµν⟩BD =−Cgeom H4gµν.(162) By Theorem C.1, the universal thermodynamic contribution fixed by (TdS, SdS, Vq) is ⟨Tmod µν ⟩=−Cmod H2M2 Plgµν.(163) Summing the two contributions produces the combined effective stress tensor (161). □ 49 Interpretation. Equation (161) shows that every admissible observer qreconstructs a vacuum stress tensor consisting of two conceptually distinct terms: •an H4ultraviolet contribution from QFT coarse–graining, determined solely by the engineering dimension of Tµν and the single coarse–graining scale ℓq∝H−1; •an H2M2 Pl infrared contribution tied to static–patch horizon thermodynamics. Both contributions are separately covariant, separately conserved within the static patch, and vanish in the flat–space limit H→0. Role in the global (G,X)–structure. The local form (161) is the sole input needed for the global consistency analysis of Sec. D. In the (G,X) picture, each observer–patch Dq carries a locally reconstructed stress tensor of this universal form, and global consistency requires the collection {T(q) µν }to satisfy appropriate overlap and cocycle conditions. The universal (H4, H2M2 Pl) scaling is therefore a structural constraint on the admissible RoR geometric structures. C.7 Transition to the Global (G, X)–Structure Analysis With the combined local result ⟨T(q) µν (x)⟩BD =−Cgeom H4+Cmod H2M2 Pl gµν(x), Cgeom, Cmod =O(1),(164) we are now in position to reinterpret the Relativity of Reconstruction (RoR) framework in geometric terms. Local model space X.For each static–patch observer q, the reconstructed stress tensor (164) lies in the two–parameter family X≡Tµν =−a H4+b H2M2 Plgµν a, b ∈R, a, b =O(1),(165) which we call the model space for observer–relative reconstruction. Each observer therefore provides a chart ψq:Dq−→ X, x 7→ T(q) µν (x).(166) Structure group G.Changes of observer correspond to maps gq→q′:X→X(167) induced by: 1. static–patch isometries (geometric transport), 2. modular flow associated with horizon thermality, 3. changes in coarse–graining scale under boosts or redshift. These transformations preserve the (H4, H2M2 Pl) functional form and act linearly on the coefficient pair (a, b). The admissible set of such transformations forms the structure group G⊂GL(2,R).(168) 50 Local-to-global problem. We may therefore regard the collection of observer reconstructions {ψq:Dq→X}as an atlas on the spacetime manifold, with transition functions gq→q′∈Gdefined on the overlaps Dq∩Dq′. The global RoR question is: Does the family {T(q) µν }assemble into a globally consistent (G, X)–structure on de Sitter spacetime, and under what conditions is this structure unique? In other words, the observer–relative stress tensors play the role of local coordinate charts valued in the model space X, and the consistency requirements of overlapping patches correspond to the cocycle conditions gq→q′′ =gq′→q′′ ◦gq→q′on Dq∩Dq′∩Dq′′ .(169) Next step: the Thurston analysis. The remainder of the paper (Sec. D) develops the geometric viewpoint: identifying •the precise subgroup Gof allowed transitions, •the resulting (G, X)–structure on de Sitter, •obstructions to global extension, •uniqueness or moduli of compatible RoR structures, •and the implications for semiclassical Einstein equations. This geometric framework is the natural home for understanding how observer–dependent reconstructions assemble into a coherent global picture. D Global RoR Geometry as a (G, X)–Structure The local reconstruction result of Sec. C.6 shows that every static–patch observer qassigns to the Bunch–Davies vacuum an effective stress tensor of the universal form ⟨T(q) µν ⟩=−c1(q)H4+c2(q)H2M2 Plgµν, c1(q), c2(q) = O(1). The pair c1(q), c2(q)contains the complete local observer–relative information reconstructed by Fq. We now reinterpret this family of local reconstructions in Thurston’s language of (G, X)–structures. D.1 Model Space X We define the model space X∼ =R2, with global coordinates (c1, c2) representing the coefficients of the geometric (H4) and modular (H2M2 Pl) contributions to the observer–relative effective stress tensor. 51 Each observer qtherefore determines a single point ψq:= c1(q), c2(q)∈X. Physically admissible points lie in a region of Xdetermined by positivity constraints, de Sitter covariance, and the flat–space limit, but for the purpose of defining the geometric structure we take the full space X=R2as the model space. D.2 Structure Group G When passing from one observer qto another q′, the reconstructed coefficients (c1, c2) transform according to the physical mechanisms governing observer change: 1. Static–patch isometries, transporting the stress tensor covariantly across overlaps and inducing a linear action on (c1, c2). 2. Modular-flow rescalings, acting homogeneously on the modular coefficient c2through the universal thermodynamic structure of the de Sitter horizon. 3. Redshift and boost dependence of the coarse–graining scale ℓq∝H−1, modifying the geometric coefficient c1(linearly at leading order in the quotient construction). These operations act linearly on (c1, c2), so the transition maps gq→q′:X→X form a subgroup G⊂GL(2,R). In coefficient form the transformation law is c1(q′) c2(q′)!=gq→q′ c1(q) c2(q)!, gq→q′∈G. Physical requirements impose strong restrictions on G: •the flat–space limit requires gq→q′→⊮as H→0; •positivity and normalization of the filter Fqimply that physically realized coefficients satisfy c2≥0 (although the model space itself remains X=R2); •de Sitter covariance suggests that, in an appropriate basis, elements of Gare upper– triangular or diagonal. Thus the RoR construction naturally defines a (G, X)–structure with local charts ψq: Dq→Xand transition functions gq→q′relating overlapping observer patches. D.3 The Observer Atlas We now describe the family of static–patch observers as an atlas suitable for defining a (G, X)–structure. 52 Observer space. Let Odenote the (possibly infinite) collection of admissible inertial static–patch observers in de Sitter spacetime. Each observer q∈ O has an associated causal diamond (the static patch) Dq⊂dS4, an open region on which the observer has full access to local operators and horizon thermodynamic data. Atlas of patches. The collection U={Dqq∈ O } forms an open cover of dS4. Distinct observers generally have overlapping domains Dq∩Dq′=∅, and these overlaps will support the transition functions of the RoR (G, X)–structure. Chart maps. For each observer q, the reconstruction operator Fqassigns to the Bunch– Davies vacuum a pair of coefficients ψq(x) = c1(q), c2(q)∈X∼ =R2, x ∈Dq. Although formally defined pointwise, de Sitter invariance implies that the map ψq:Dq→X is constant on its domain. Constancy within each chart. Since the reconstructed stress tensor satisfies ⟨T(q) µν (x)⟩=−c1(q)H4+c2(q)H2M2 Plgµν(x), the coefficients (c1, c2) are constant throughout the static patch. Thus ψq(Dq)={(c1(q), c2(q))} ⊂ X. Geometrically, each observer patch is labeled by a single point in model space X, representing the local RoR data rather than providing spacetime coordinates. Interpretation. The atlas Utogether with the chart maps {ψq}q∈O plays the role of a Thurston atlas: each static–patch observer provides a local description of the “RoR geometry” valued in X. Transition maps on overlaps, introduced in the next subsection, will determine whether the family {ψq}assembles into a globally consistent (G, X)–structure on dS4. D.4 Transition Maps and the RoR Cocycle Condition Given two observers qand q′whose static patches overlap, Dq∩Dq′=∅, we define the transition map gq→q′:X→X, and show that these maps satisfy the Thurston cocycle condition. 53 Definition of the transition map. On the overlap Dq∩Dq′, both observers reconstruct an effective stress tensor of the form ⟨T(q) µν ⟩=−c1(q)H4+c2(q)H2M2 Plgµν, ⟨T(q′) µν ⟩=−c1(q′)H4+c2(q′)H2M2 Plgµν. Because both reconstructions refer to the same underlying physical state in the same spacetime region, the pairs of coefficients must be related by a linear transformation. This linearity follows from the facts that: •the reconstruction Fqis affine-linear in the stress tensor, and •the coefficients (c1, c2) appear only as the weights of the two invariant basis tensors {H4gµν, H2M2 Plgµν}. Thus every admissible observer-change acts linearly on the coordinate pair (c1, c2): c1(q′) c2(q′)!=gq→q′ c1(q) c2(q)!, gq→q′∈G⊂GL(2,R). The transition map gq→q′is defined on the overlap Dq∩Dq′. Since both chart maps are constant on their domains, the definition is well-posed and independent of the point of evaluation in the overlap. Why the transition maps are constant matrices. Three ingredients imply constancy of gq→q′on each connected overlap: 1. BD invariance. The Bunch–Davies vacuum is de Sitter invariant; therefore the reconstructed stress tensor has the same algebraic form at every point of Dq∩Dq′. If the state were not de Sitter invariant, the transition map could become point-dependent. 2. Constancy of chart coordinates. The coefficients (c1(q), c2(q)) assigned by any observer are constant throughout Dq. 3. Covariance and locality. The relative transformation depends only on the global frame relationship between qand q′(isometry, redshift, modular normalization), not on any local spacetime position. Thus the map depends only on the relative observer data, not on x. Hence gq→q′is represented by a constant 2 ×2 matrix on each connected component of the overlap. 54 The cocycle condition. For three observers q,q′, and q′′ with triple overlap Dq∩Dq′∩ Dq′′ =∅, global consistency demands that transforming from qto q′′ directly is equivalent to passing through q′: gq→q′′ =gq′→q′′ ◦gq→q′. Indeed, c1(q′′) c2(q′′)=gq→q′′ c1(q) c2(q), but also, c1(q′′) c2(q′′)=gq′→q′′ c1(q′) c2(q′)=gq′→q′′ gq→q′c1(q) c2(q). Hence the equality of matrices follows. Thus the assignment q7→ ψqdefines a representation of observer changes into the structure group G. Holonomy and the global obstruction to a single vacuum energy. For a loop of observers q0→q1→···→qn=q0, the composed transition map gγ:= gqn−1→q0···gq1→q2gq0→q1∈G is the holonomy of RoR around the loop γ. •If gγ=⊮for all loops, a single global pair (c1, c2) exists and the reconstruction is globally trivial. •If gγ=⊮for some loop, then although the BD vacuum is globally well-defined, its reconstruction is not globally consistent across all observers. This is the RoR mechanism that obstructs the existence of a single global renormalized stress tensor. D.5 The Developing Map and Holonomy Representation With the transition maps {gq→q′}and the cocycle condition established, we now construct the developing map and the associated holonomy representation. These two objects are the global invariants of a (G, X)–structure and capture precisely how local observer reconstructions fail to assemble into a single global stress tensor. The developing map assigns to each point in the universal cover of de Sitter spacetime a point in the model space X, encoding the observer–relative vacuum data along lifted paths. The holonomy representation records how the developing map changes under closed loops of observers; nontrivial holonomy is the mathematical expression of the RoR principle that the vacuum energy is not a globally defined quantity but depends on the relational choice of observer. We now construct these objects explicitly. 55 Orientation. The developing map records how the locally reconstructed parameters (c1, c2) are transported along paths in the observer atlas. It plays the same role as the developing map of a classical (G, X)–structure. Constancy on patches. Since (c1(q), c2(q)) are constant on each static patch Dq, the value ψq0(x0) is independent of the choice of basepoint x0∈Dq0. This ensures that patchwise continuation is well-defined. Universal cover. Let f dS4denote the universal cover. Although dS4is already simply connected, using the universal cover removes any possible monodromy arising from loops in the observer atlas rather than from spacetime topology. Holonomy is therefore captured purely by the transition maps in G. Continuation across overlaps. Choose a lift e Dq0of a static patch and fix a point ˜x0∈ e Dq0. We extend the chart map ψq0to all of f dS4by continuation across overlapping lifted patches using the transition maps. If x∈e Dqkand e Dq0,e Dq1, . . . , e Dqk is any chain of overlapping lifts connecting ˜x0to x, then we define dev(x) := gq0→q1gq1→q2· · · gqk−1→qkψq0(˜x0). Well-definedness. Because the transition maps satisfy the cocycle condition gq→q′′ =gq′→q′′ ◦gq→q′, the above definition is independent of the chosen chain of overlaps. Hence: dev : f dS4−→ X is a well-defined smooth map: the developing map of the RoR (G, X)–structure. Observer loops. Although dS4is topologically simply connected, nontrivial holonomy arises from loops in the observer atlas, i.e. sequences of overlapping static patches that return to the initial patch. These observer loops generate an effective fundamental group for the (G, X)–structure. Definition of holonomy. Let q0→q1→···→qn=q0 be an observer loop. The composed transition map gγ:= gqn−1→q0···gq1→q2gq0→q1∈G is the holonomy of the loop γ. 56 Holonomy representation. These holonomies define a group homomorphism Hol : π1(O)−→ G, where π1(O) is the loop space generated by overlapping observer patches. The developing map satisfies the equivariance relation dev(γ·˜x) = Hol(γ)·dev(˜x), exactly as in classical (G, X)–structures. Meaning of the developing map. The map dev : f dS4→X encodes the globally continued observer-relative parameters (c1, c2). It is the RoR analogue of the geometric developing map: the global extension of all local reconstructions into the model space X. Meaning of holonomy. If Hol(γ) = ⊮for every observer loop, then dev descends to a globally well-defined map dS4→X, and all observers agree on a single global pair (c1, c2). If Hol(γ)=⊮for some loop, then no globally consistent pair (c1, c2) exists. Although the BD vacuum is globally well-defined, its reconstruction is not globally consistent across all observers. Nontrivial holonomy therefore forbids any global, observer-independent functional ⟨Tµν⟩, violating precisely the globality assumption entering Weinberg’s no-go theorem. D.6 Global RoR Structure Theorem We now combine the local reconstruction data, the transition maps, the cocycle relations, and the developing-map construction into a single global statement. Let U={Dq}q∈O be the open cover of dS4by static–patch observer domains. For each q, let ψq:Dq→X∼ =R2be the constant chart map ψq(x)=(c1(q), c2(q)), with ⟨T(q) µν ⟩=−c1(q)H4+c2(q)H2M2 Plgµν. Let G⊂GL(2,R) be the structure group generated by static–patch isometries, redshift– induced geometric rescalings, and modular flow transformations. Let Π1(U) denote the observer-loop group: the set of equivalence classes of loops Dq0→Dq1→···→Dqn=Dq0, formed by successions of overlaps in the cover U. Then: (1) The data {(Dq, ψq)}with transition maps gq→q′∈Gdefined on each overlap Dq∩Dq′ forms a well-defined (G, X)–structure on the observer atlas. 57 ROR_GX_structure_schematic.pdf Figure 1: Schematic representation of the RoR (G, X)–structure. Each static–patch observer qcovers a domain Dq⊂dS4on which the reconstructed coefficients (c1(q), c2(q)) are constant, defining a chart map ψq:Dq→X∼ =R2. On overlaps Dq∩Dq′, consistency of the reconstructions forces the two charts to be related by a transition map gq→q′∈G⊂GL(2,R), acting linearly on the pair (c1, c2). A succession of overlaps Dq0→Dq1→···→Dqn=Dq0 forms an observer loop, whose associated product of transition maps is the holonomy element Hol(γ) = gqn−1→qn···gq0→q1∈G. Lifting the observer atlas to the universal cover f dS4 eliminates monodromy and yields a globally defined developing map dev : f dS4→Xsatisfying the equivariance relation dev(γ·˜x) = Hol(γ)·dev(˜x). Nontrivial holonomy is equivalent to the nonexistence of a globally defined, observer–independent renormalized stress tensor, and therefore provides the geometric mechanism by which the RoR framework avoids the globality assumption underlying Weinberg’s no–go theorem. 64